Unitary representations of cyclotomic Hecke algebras at roots of unity: combinatorial classification and BGG resolutions

We relate the classes of unitary and calibrated representations of cyclotomic Hecke algebras and, in particular, we show that for the most important deformation parameters these two classes coincide. We classify these representations in terms of both multipartition combinatorics and as the points in the fundamental alcove under the action of an affine Weyl group. Finally, we cohomologically construct these modules via BGG resolutions.


Introduction
Unitary representations play a fundamental role in the representation theory of real, complex, and p-adic reductive groups [ALTV20, Vog00, EH04, BC20].Unitary representations are often the most important representations appearing "in nature" via quantum mechanics [Wig39] and harmonic analysis [Mac92].Furthermore, they tend to admit nice structural and homological properties, such as explicit eigenbases and resolutions by Verma modules.
In this paper, we study unitary representations of a family of infinite discrete groups: the affine braid groups.These are groups B ea n of n braids on the cylinder, see [GTW17], and project onto the usual Artin braid groups by "flattening" the cylinder.Of course, the representation theory of the group B ea n is extremely complicated and the problem would be intractable without imposing certain conditions on our representations.The condition we impose is that our representations factor through an affine Hecke quotient of the group algebra CB ea , that is, the following skein-like relation is satisfied ( †) (T i − q)(T i + 1) = 0 for some q ∈ C × and every i = 1, . . ., n − 1, where T i is the overcrossing of the i-th and (i + 1)st strands.The algebra H aff q (n) := CB ea n /(T i − q)(T i + 1) is known as the affine Hecke algebra.Besides being interesting in and of itself, the algebra H aff q (n) appears in the theory of knot invariants, categorification, and the representation theory of p-adic reductive groups.Let us now discuss our methods and results in more detail.
First main result.First, we relate the unitary representations of H aff q (n) to the class of calibrated representations.These are a well-studied class of representations of H aff q (n) that are defined by the condition that the Jucys-Murphy subalgebra A ⊆ H aff q (n) acts semisimply, see [Ram03b,Ram03a,Ruf06,Kle96].Calibrated representations exhibit many of the properties that make unitary representations interesting (for example, by definition they come equipped with an A-eigenbasis, which is unique if the representation is simple) and it easily follows that, in fact, every unitary representation is calibrated.The converse is of course not true, but it turns out that if we restrict to certain representations (in a sense, the most complicated ones) then the story changes.
To be more precise, every irreducible representation of H aff q (n) factors through a cyclotomic quotient, H q,Q 1 ,...,Q ℓ (n) which depends on several parameters q, Q 1 , . . ., Q ℓ .The representation theory of H q,Q 1 ,...,Q ℓ (n) is most interesting when we specialise the parameter q to be a root of unity and the parameters Q i = q s i for 1 ≤ i ≤ ℓ.
Theorem A. Let e > 1 and s 1 , . . ., s ℓ be integers.Let q = exp(2π √ −1/e) and Q i = q s i .A representation of the algebra H q,Q 1 ,...,Q ℓ is unitary if and only if it is calibrated.
Second main result.Next, we use Theorem A to combinatorially classify the unitary representations.Given an integer e > 1 and s = (s 1 , . . ., s ℓ ) ∈ Z ℓ a charge, we denote the algebra H q,Q 1 ,...,Q ℓ (n) as above simply by H s (n).The definition of the algebra H s (n) depends only on the reduction modulo e of the charge s, so we can assume that s 1 ≤ s 2 ≤ . . .≤ s ℓ < s 1 + e.Our choice of charge allows us to provide a particularly simple classification of unitary modules in terms of multipartition combinatorics: Theorem B. Let q = exp(2π √ −1/e) and fix a charge s = (s 1 , . . .s ℓ ) such that s 1 ≤ s 2 ≤ . . .≤ s ℓ < s 1 + e.The simple H aff q (n)-module D s (λ) is unitary if and only if the following equivalent conditions hold: • λ is cylindric, its right border strip has period at most e and its reading word is increasing; • λ ∈ F h the fundamental alcove under an s-shifted action of an affine Weyl group of type A; • D s (λ) is calibrated.
Precise definitions for the terminology in the first two conditions are given in Subsections 3.1 and 5.1.
Theorem B gives the first classification of calibrated representations for the algebra H s (n) in terms of Young diagrams of multipartitions.Note that other combinatorial classifications in terms of weights and skew-Young diagrams are given in [Ram03a,Ram03b].Theorem B can be seen as the analogue of [Jac04, Theorem 4.1] for calibrated representations -both results provide the first closed-form description of the given family of irreducible modules in terms of multipartitions.
Third main result.The combinatorial description of the unitary representations in Theorem B leads to our third main result, a multiplicity-free character formula for these representations and their cohomological construction by way of BGG resolutions.
Over C[q, Q 1 , Q 2 , . . ., Q ℓ ], the algebra H q,Q 1 ,...,Q ℓ (n) is semisimple and the simple Specht modules S(µ) are indexed by the set of ℓ-multipartitions µ of n.For e > 1 and s ∈ Z ℓ one can place a corresponding integral lattice on each Specht module S(µ), obtaining a family of (non-semisimple) H s (n)-modules S s (µ) by specialisation of the parameters q = exp(2π √ −1/e) and Q i = q s i .Our choice of s ∈ Z ℓ allows us to construct the unitary simple D s (λ) as the head of the Specht module S s (λ) for λ as in Theorem B. A BGG resolution of D s (λ) is a resolution of D s (λ) by a complex whose terms are direct sums of Specht modules S s (µ).
Given a unitary simple module D s (λ), we consider the set of multipartitions µ dominating λ and having the same residue multiset as λ (with respect to the charge s).For such µ we write µ λ, see Section 1.The affine symmetric group S h , where h is the number of rows of λ, acts naturally on this set of multipartitions, endowing it with the structure of a graded poset in which λ is the unique element of length 0. We then construct a BGG resolution of D s (λ) as follows: Theorem C. Associated to each unitary simple module, D s (λ), we have a complex C • (λ) = µ λ S s (µ) ℓ(µ) with differential given by an alternating sum over all "simple reflection homomorphisms".This complex is exact except in degree zero, where H 0 (C • (λ)) = D s (λ).The underlying graded character is as follows, Moreover, the module D s (λ) admits a characteristic-free basis {c S ⊗ Z k | S ∈ Path F h (λ)} where Path F h (λ) ⊆ Path h (λ) is the subset of paths which never leave the fundamental alcove F h .In the case of the unitary representations of the Hecke algebra of the symmetric group, the resolutions of Theorem C were the subject of the authors' previous work [BNS18].Theorem C vastly generalises this work to all unitary representations of all cyclotomic Hecke algebras.
Aspects of this story should be very familiar to the experts: we have an algebraic object (in this case a cyclotomic or affine Hecke algebra) for which there exists a "nice family" of irreducible representations (in this case, the unitary representations) which can be combinatorially classified and constructed via explicit bases and BGG resolutions.Analogous stories exist for ladder representations of p-adic groups [BC15], finite dimensional representations of (Kac-Moody) Lie algebras [BGG75,KK79], and homogeneous representations of antispherical Hecke categories [BHN20].
We remark that while the definition of a unitary module depends crucially on the ground field being C, the condition to be calibrated makes sense for arbitrary fields.In this manner, Theorems B and C both admit characteristic-free generalisations which are proven in this paper.
Regular p-Kazhdan-Lusztig theory.The coefficients in equation ( † †) are equal to "regular" (or "non-singular") inverse (p-)Kazhdan-Lusztig polynomials.In fact, the proof of Theorem B involves passing from the cyclotomic Hecke algebras to the setting of Elias-Williamson's diagrammatic category for "regular" Soergel-bimodules.For the general linear group, GL h , the "regular" Soergel-bimodules control the representation theory of the principal block (the block containing the trivial representation k = ∆(k h ) for n = kh) if and only if p > h.On the other side of Schur-Weyl duality this means that "regular" Soergel-bimodules control the representation theory of the Serre subcategory of kS n -mod corresponding to the poset {λ | λ (k h ) for n = kh} and we remark that the simple kS n -module labelled by the partition (k h ) is calibrated providing p > h.
For higher levels ℓ > 1, one can ask "to what extent is the cyclotomic Hecke algebra controlled by regular (p-)Kazhdan-Lusztig theory?".Of course, the Schur-Weyl duality with the general linear group no longer exists.However, one can speak of calibrated representations of the cyclotomic Hecke algebra.In fact, the largest Serre subcategory of (a block of) the cyclotomic Hecke algebra controlled by regular p-Kazhdan-Lusztig theory is given by the poset {µ | µ λ} where D s (λ) is the minimal calibrated simple module in the block (under the order ).Thus the Serre quotients carved out by calibrated representations of cyclotomic Hecke algebras play the same role as that of principal blocks of algebraic groups for p > h.
Structure of the paper.Section 1 introduces the combinatorics that will play an important role in this paper.Then we study unitary representations in Section 2 where we prove Theorem A, see Theorem 2.21.Sections 3 and 4 are devoted to the proof of Theorem B, which involves intricate combinatorial constructions.In Section 5 we recall previous work of the first author together with A. Cox and A. Hazi [BCH20] that will allow us to prove Theorem C. We do this in Section 6.In this section we also discuss the consequences of our work in the representation theory of rational Cherednik algebras.Finally, in Appendix A we use our techniques to give a complete classification of unitary representations of the Hecke algebra of the symmetric group.While this has mostly appeared in the literature, see [Sto09], we believe it gives a good feeling for the usage of calibrated representations in this setting, and corrects an oversight of [Sto09].
Definition 1.1.We call an ℓ-tuple of integers s = (s 1 , s 2 , . . ., s ℓ ) ∈ Z ℓ an ℓ-charge or simply a charge.Given e ∈ Z ≥2 , we say that s is cylindrical if We define a composition λ of n to be a finite sequence of non-negative integers (λ 1 , λ 2 , . ..) and whose sum |λ| = λ 1 + λ 2 + . . .equals n.We say that λ is a partition if, in addition, this sequence is weakly decreasing.We let λ t denote the transpose partition.An ℓ-multicomposition (respectively ℓ-multipartition or simply ℓ-partition) λ = (λ 1 , λ 2 , . . ., λ ℓ ) of n is an ℓ-tuple of compositions (respectively partitions) such that |λ| := |λ 1 | + |λ 2 | + . . .+ |λ ℓ | = n.We will denote the set of ℓ-multicompositions (respectively ℓ-partitions) of n by C ℓ (n) (respectively by P ℓ (n)).Given λ = (λ 1 , λ 2 , . . ., λ ℓ ) ∈ P ℓ (n), the Young diagram of λ is defined to be the set of boxes (or nodes), We do not distinguish between the multipartition and its Young diagram.We draw the Young diagram of a partition by letting c increase from left to right and r increase from top to bottom.We refer to a box (r, c, m) as being in the rth row and cth column of the mth component of λ.We draw the Young diagram of a multipartition by placing the Young diagrams of λ 1 , . . ., λ ℓ side by side from left to right as m runs from 1 to ℓ.Finally, a tableau T on a multipartition λ is a bijection from the set of boxes of λ to {1, 2, . . ., |λ|}.The tableau T is called standard if it is increasing along the rows and columns of each component.We let Std(λ) denote the set of all standard tableaux on λ.If T ∈ Std(λ) is a standard tableau, then Shape(T↓ {1,...,k} ) is the multipartition whose Young diagram consists of all the boxes with labels ≤ k.Finally, we denote by ∅ the empty multipartition.
Note that the residue of a box in λ depends on the choice of the charge s ∈ Z ℓ .For a tableau T on λ, we let res(T) denote the residue sequence consisting of res(T −1 (k)) for k = 1, . . ., n in order.
Definition 1.6.Given a partition λ = (λ 1 , . . ., λ h ) such that λ h > 0, we set h(λ) to be the height of the partition, that is h(λ) = h.Given a multipartition λ = (λ 1 , λ 2 , . . ., λ ℓ ), we set h(λ) = (h 1 , . . ., h ℓ ) to be the ℓ-composition formed of the heights of the component partitions.Given λ = (λ 1 , λ 2 , . . ., λ ℓ ), we define the height of the multipartition to be the integer and with at least one of these inequalities being strict.We refer to any value 1 ≤ m ≤ ℓ for which the inequality is strict as a step change.For an s-admissible h ∈ N ℓ , we let P h (n) denote the set of all λ ∈ P ℓ (n) such that h(λ) = h.Definition 1.8.Let h ∈ N ℓ be s-admissible and let 1 ≤ m ≤ ℓ.Given λ ∈ P h (n), we define the reverse column reading tableau, T (m,λ) , to be the standard tableau obtained by filling the first column of the (m − 1)th component then the first column of the (m − 2)th component and so on until finally filling the first column of the mth component and then repeating this procedure on the second columns, the third columns, etc.
Given λ ∈ P h (n), we let Add h (λ) (respectively Rem h (λ)) denote the set of all addable (respectively removable) boxes of the Young diagram so that the resulting Young diagram is the Young diagram of a multipartition belonging to P h (n + 1) (respectively P h (n − 1)).We let Add i h (λ) (respectively Rem i h (λ)) denote the subset of nodes of residue equal to i ∈ Z/eZ.Dropping the subscript h we obtain the usual sets of addable and removable i-nodes of a multipartition.

1.2.
The sl e -crystal.Fix e ≥ 2 and a charge s ∈ Z ℓ .The sl e -crystal is a simply directed graph on the set of vertices consisting of all ℓ-partitions.Its arrows are given by a rule for adding at most one box of each residue i ∈ Z/eZ to a given ℓ-partition.We define the crystal operators ẽi and fi , which remove and add (respectively) at most one box of residue i. Definition 1.12.[FLO+99, Theorem 2.8] Fix i ∈ Z/eZ, s ∈ Z ℓ , and an ℓ-partition λ.
• Form the i-word of λ by listing all addable and removable i-boxes of λ in increasing order from left to right (according to ⊲ s ) and then replacing each addable box in the list by the symbol + and each removable box by the symbol −. • Next, find the reduced i-word of λ by recursively canceling all adjacent pairs (−+) in the i-word of λ.The reduced i-word is of the form (+) a (−) b for some a, b ∈ Z ≥0 .• Define fi as the operator that adds the addable i-box to λ corresponding to the rightmost + in the reduced i-word of λ.If the reduced i-word of λ contains no + then we declare that fi λ = 0. Likewise, define ẽi as the operator that removes the removable i-box from λ corresponding to the leftmost − in the reduced i-word of λ.If the reduced i-word of λ contains no − then we declare that ẽi λ = 0.
The directed graph with vertices all ℓ-partitions and arrows λ i → µ if and only if µ = fi λ, i ∈ Z/eZ, is called the sl e -crystal.We have fi µ = λ if and only if ẽi λ = µ.Definition 1.13.The i-box that is added by fi , if it exists, is called a good addable i-box of λ.Similarly, the box that is removed by ẽi is called a good removable i-box of λ.
Remark 1.14.Given i ∈ Z/eZ, if λ has only one removable i-box b and no addable i-box of charged content greater than or equal to co s (b), then ẽi (λ) = λ \ {b}.We will use this observation without further mention.
The sl e -crystal is in general disconnected and we will be interested in its connected component containing the empty multipartition ∅.The vertices in this connected component of the sl e -crystal do not, in general, admit closed formulas (one must instead search for a sequence of good nodes by repeated use of Definition 1.12).However, for our choice of cylindric charge s ∈ Z ℓ (as in Definition 1.1) we have the following combinatorial description from [FLO+99].

Unitary representations
2.1.The affine Hecke algebra.Let us recall that the (extended) affine braid group, B ea n has generators T 1 , . . ., T n−1 , x 1 , . . ., x n subject to the following relations: We may think of B ea n as the braid group on the cylinder.The element T i correspond to the usual braid generator that crosses two adjacent strands, and the element x i corresponds to looping the i-th strand around the cylinder so that it comes back to the i-th position, see Figure 1 below.
The braid generators x i and T i .We remark that these are braids on a cylinder, so the left and right-sides of the rectangles are to be identified.
The affine Hecke algebra is the quotient of the group algebra of B ea n by a skein-type relation: Definition 2.1.Let R be a domain and q ∈ R × , q = ±1.The (extended) affine Hecke algebra H aff q (n, R) is the quotient of the group algebra RB ea n by the relations When the domain R is clear from the outset, we will simply denote the affine Hecke algebra by H aff q (n).
Remark 2.2.It is customary to define the elements X i := q 1−i x i , so that in H aff q (n) we have the relation T i X i T i = qX i+1 .
Remark 2.3.The finite Hecke algebra H q (n) can be realized as a subalgebra of H aff q (n) generated by T 1 , . . ., T n−1 .It can also be realized as the quotient H aff q (n)/(X 1 − 1).This is akin to the finite braid group B n being both a subgroup and a quotient group of B ea n .
Remark 2.4.The elements X 1 , . . ., X n are known as the Jucys-Murphy elements of H aff q (n).We will call the algebra A := R[X ±1 1 , . . ., X ±1 n ] ⊆ H aff q (n) the Jucys-Murphy subalgebra.It is isomorphic to the algebra of Laurent polynomials in n variables and we have a vector space decomposition When R is a domain of characteristic zero it is known, see e.g.[CG10, Proposition 7.1.14],that the center of H aff q (n) is Z(H aff q (n)) = A Sn , the algebra of symmetric Laurent polynomials in the Jucys-Murphy elements.Thus, H aff q (n) is finite over its center and, when R is a field F of characteristic zero, every irreducible representation of H aff q (n) is finite-dimensional.If moreover F is algebraically closed it follows, looking at the eigenvalues of X 1 , that every irreducible representation of H aff q (n) factors through an algebra of the form This is known as the cyclotomic Hecke algebra, or the Ariki-Koike algebra.It is a finite-dimensional F-algebra, of dimension precisely n!ℓ n , [AK94].
2.2.Unitary representations.For this subsection, we let R = C, the complex field.We make the convention that a Hermitian form on a finite-dimensional complex vector space is linear on the first variable and conjugate-linear on the second.Recall that we have defined the affine Hecke algebra H aff q (n) as a quotient of the group algebra CB ea n , so the following notion makes sense.
Definition 2.5.We say that a finite-dimensional H aff q (n)-representation is unitary if it admits a positive-definite B ea n -invariant Hermitian form.
Let us remark that not all affine Hecke algebras admit nontrivial unitary representations, where by non-trivial we mean that at least one T i does not act by −1.Indeed, the parameter q plays an essential role in here.
Lemma 2.6.Assume that H aff q (n) admits a nontrivial unitary representation.Then, q ∈ C × lies in the unit circle.
Proof.Let M be a nontrivial unitary H aff q (n)-representation.Let m ∈ M be an eigenvector for T 1 with eigenvalue q, which we know exists because M is nontrivial.Then so that q = q −1 .Thus, q is in the unit circle.
Remark 2.7.We remark that Lemma 2.6 follows from the following more general result.Let G be any group and let M be a finite-dimensional unitary representation of G.Then, every eigenvalue of g on M lies in the unit circle, for any g ∈ G.Note that this statement is obvious when G is a finite group as in this case every eigenvalue of g is, in fact, a root of unity.The proof in the general case is just as that of Lemma 2.6.
Since every cyclotomic Hecke algebra H q,Q 1 ,...,Q ℓ (n) is a quotient of the affine Hecke algebra it makes sense to speak about unitary representations of H q,Q 1 ,...,Q ℓ (n).Just as in Lemma 2.6, we have that if H q,Q 1 ,...,Q ℓ (n) admits a unitary representation then all complex numbers q, Q 1 , . . ., Q ℓ ∈ C × must lie in the unit circle.
Let us remark that the cyclotomic Hecke algebra H q,Q 1 ,...,Q ℓ (n) is a quotient of the group algebra of a different braid group: the braid group B(ℓ, 1, n), which is defined as follows.Let G(ℓ, 1, n) := S n ⋉ (Z/ℓZ) n be the cyclotomic group.If we think of this group as the group of n × n permutation matrices whose nonzero entries are ℓ-roots of unity, we get an action of G(ℓ, 1, n) on h := C n .Let h reg be the locus where this action is free, it can be shown that this is the complement of a hyperplane arrangement in C n .Then, For example, when ℓ = 1, the group B(1, 1, n) is the usual Artin braid group on n strands.It is clear from the definitions, see e.g.[BMR98], that a H q,Q 1 ,...,Q ℓ (n)-representation is unitary if and only if it admits a positive-definite B(ℓ, 1, n)-invariant Hermitian form.On the other hand, Rouquier has shown using the representation theory of rational Cherednik algebras, [SA20, Proposition 4.5.4], that every irreducible H q,Q 1 ,...,Q ℓ (n)-representation admits a (unique up to R × -scalars) non-degenerate B(ℓ, 1, n)-invariant Hermitian form.Thus, the question of unitaricity is that of positive-definiteness of this form.
2.3.Unitary representations via * -products.In [BC20], Barbasch and Ciubotaru define a class of representations that are closely related to the unitary representations we study in this paper.The goal of this section is to explore this relation.In order to do so, throughout this subsection R = C[q, q −1 ] and H aff q (n) := H aff q (n, R).
Definition 2.8.A * -product on H aff q (n) is a conjugate-linear, involutive antiautomorphism of H aff q (n).Given a * -product on H aff q (n), a representation M of H aff q (n) is called * -unitary if it admits a positive-definite Hermitian form which is H aff q (n)-invariant in the sense that In [BC20], the authors define * -products on H aff q (n) and study the notion of unitary representations for these * -products.These are the first two * -products of the following proposition.
Proposition 2.9.The following formulas define * -products on H aff q (n).(1) Proof.For • and ⋆, see [BC20, Definition 2.3.1]..For †, note that the relation (T i + 1)(T i − q) = 0 is equivalent to (T −1 i + 1)(T −1 i − q −1 ) = 0, so this is preserved under †.It is also clear that † preserves the relations among the X i , as well as the braid relations among the T i .Finally, we have which finishes the proof.
It is clear that a representation M of H aff q (n) is unitary in the sense of Definition 2.5 if and only if it is †-unitary in the sense of [BC20].Representations that are ⋆-unitary and •-unitary have been studied in loc.cit., where they are shown to be related to unitary representations of general linear groups over p-adic fields.Note that both • and ⋆ commute with the action of q, while † does not.Nevertheless, the * -products • and † are closely related, as witnessed by the following result.
Proposition 2.11.The map ι defined on generators by ι(T Remark 2.12.From the formulas, it may seem that ι and † coincide.Note, however, that ι is an automorphism while † is an anti -automorphism.Also, ι is C-linear while † is conjugate-linear. Proof.That ι defines a C-linear involutive algebra automorphism is easy to check from the relations on H aff q (n).Since ι is C-linear while † is conjugate linear, both compositions ι • † and † • ι are conjugate linear.Also, since ι is an automorphism while † is an antiautomorphism, both ι • † and † • ι are antiautomorphisms.Thus, it suffices to check the equality ι We remark, however, that we do not obtain an equivalence between categories of †-unitary and •-unitary representations.To do this, we must find an automorphism ϕ of It is unlikely that such an automorphism exists, at least in the algebraic setting.Roughly speaking, if an irreducible representation is †-unitary then we must have that q acts by a complex number of norm 1 while, if a representation is •-unitary then q must act by a real number.It is possible that one may obtain equivalences between †-unitary and •-unitary representations by working with representations of the formal affine Hecke algebra, [HMLSZ14] so that we can take exponential and logarithms of the variable (that should be thought of as log(q)), but we do not do it here.
2.4.Unitary representations are calibrated.We go back to the setting of R = C.Moreover, throughout this section we let q ∈ C × be in the unit circle.Let us recall the following important notion from the representation theory of affine Hecke algebras, [Ram03b].
Definition 2.13.A H aff q (n)-representation M is called calibrated if the Jucys-Murphy subalgebra A acts semisimply on M .
Lemma 2.14.Let M be a unitary and finite-dimensional H aff q (n)-module.Then M is semisimple and calibrated.
Proof.Let N ⊆ M be an H aff q -submodule of M .It is immediate to see that the Hermitian orthogonal N ⊥ is a complement to N in M .Thus, every submodule of M splits.The proof of semisimplicity for the A-action is the same, after observing that † preserves the Jucys-Murphy subalgebra.
For a H aff q (n)-module and a vector a = (a 1 , . . ., a n ) ∈ (C × ) n , we define the a-weight space to be Thus, every unitary H aff q (n)-module is the direct sum of its weight spaces.
Lemma 2.15.Let M be unitary, with invariant Hermitian form •, • .Then, (1) Proof.Statement (1) is clear, see e.g.Remark 2.7.To prove (2), assume that both M a and M b are both nonzero, let m 1 ∈ M a , m 2 ∈ M b and i such that a i = b i .Then, Since a i = b i and, by (1), Thus, we have m 1 , m 2 = 0, as required.
2.5.When are calibrated representations unitary?The purpose of this section is to obtain necessary and sufficient conditions for a calibrated representation to be unitary.We have seen one condition a calibrated representation must satisfy in order to be unitary: all the weights appearing in it must have values in the unit circle.This necessary condition is, however, not sufficient.To obtain a complete answer we must first recall the classification of irreducible calibrated representations in terms of the weights that appear in them.This is from [Ram03b] and we follow loc.cit closely.
Since we are assuming that q = ±1 note, in particular, that if a is a calibrated weight, then a i = a i+1 , a i = a i+2 for every i.We denote by C aff ⊆ (C × ) n the set of all calibrated weights.For a calibrated weight a ∈ C aff , we say that s i = (i, i + 1) ∈ S n is an admissible transposition if a i+1 = q ±1 a i .Note that, if s i is an admissible transposition, then s i a ∈ C aff .Definition 2.17.Define the equivalence relation ∼ on C aff by saying that a ∼ b if a can be reached from b by applying a sequence of admissible transpositions.
The following result is due to Ram, [Ram03b].It can be thought of as asserting the existence of a Young seminormal form for calibrated representations.
Theorem 2.18.Let [a] ∈ C aff / ∼ be an equivalence class.Then, there exists an irreducible calibrated module M [a] whose weights are precisely [a].More precisely, M has a basis w b , b ∈ [a], and the action of H aff q (n) is given as follows: Where we define w s i b = 0 if s i is not an admissible transposition for b.Moreover, every irreducible calibrated module is of the form M [a] for a unique equivalence class [a] ∈ C aff / ∼.
We will characterize the calibrated weights a for which M [a] is indeed unitary.Since every unitary representation is semisimple, this would give us all the unitary representations of H aff q (n).First, we have seen in Lemma 2.15 that a necessary condition is that a ∈ (S 1 ) n , where S 1 ⊆ C is the unit circle.Note that Theorem 2.18 tells us that every weight space in M [a] is 1-dimensional and, as we have seen in Lemma 2.15, different weight spaces must be orthogonal under an invariant, positive-definite Hermitian form.
So let •, • be an invariant, non-degenerate Hermitian form on M [a] and assume that a ∈ (S 1 ) n .By [SA20, Proposition 4.5.4],such a form exists.By our discussion in the previous paragraph, there exist numbers and our job is to find conditions on [a] guaranteeing that all numbers A b can be chosen to have the same sign.We separate in several cases, and we make heavy use of the Young seminormal form of Theorem 2.18.
Case 1. b ′ = b, s i b.In this case, we have On the other hand, And we see that, if the form and, if the form •, • is to be positive definite, we must have is reachable from b by a sequence of admissible transpositions.We declare w b , w b = 1.By [SA20, Proposition 4.5.4],there is an invariant Hermitian form on M [a] satisfying this, and we can see that we have: We have arrived at the following result.
Theorem 2.21.An irreducible H aff q (n)-module M is unitary if and only if there exists a calibrated weight a = (a 1 , . . ., a n ) ∈ C aff satisfying the following conditions.
(2) For every b ∈ [a] and every i The following easy lemma will be useful in the future.
Lemma 2.22.Let a = (a 1 , . . ., a n ) ∈ C aff be a calibrated weight such that |a i | = 1 for every i and ℜ(a i /a j ) ≤ ℜ(q) whenever a i = a j .Then M [a] is unitary.
Proof.We need to check that the class [a] satisfies (2) of Theorem 2.21.Since every weight in [a] can be obtained by applying transpositions to a this is immediate.
We now focus on representations that factor through a fixed cyclotomic quotient of H aff q (n).
2.6.Unitary modules for the cyclotomic Hecke algebra.Now we seek to classify unitary representations of the cyclotomic Hecke algebra: .
We may and will assume that all complex numbers q, Q 1 , . . ., Q ℓ live in the unit circle.Similarly to the case of the finite Hecke algebra H q (n), we can see that if M is a calibrated representation of H q,Q 1 ,...,Q ℓ (n) and a is a weight of M , then for every i = 1, . . ., n there exist j ∈ 1, . . ., ℓ and m i ∈ Z such that a i = Q j q m i .This implies the following result, which is Theorem A from the introduction.
Lemma 2.23.Assume that q = exp(2π √ −1/e) for some e ≥ 2 and that there exist Proof.Under the assumptions of the lemma, if a is a weight of a calibrated representation then every coordinate of a is a power of q.Since ℜ(q i ) ≤ ℜ(q) unless q i = 1, the result follows from Lemma 2.22.
The parameters q, Q 1 , . . ., Q ℓ of the form appearing in Lemma 2.23 are the most interesting ones from a representation-theoretic point of view.They are those for which the representation theory of the algebra H q,Q 1 ,...,Q ℓ (n) cannot be broken-up in smaller pieces.More precisely, let us define an equivalence relation on the set of parameters Q 1 , . . ., Q ℓ by declaring Since we are interested in irreducible, unitary representations, it is enough to restrict to a single direct summand on the right-hand side of (2.24).It follows from the crystal-theoretic characterization of calibrated representations, see Theorem ) for every i.The converse, however, is not true.For example, in the most generic case when each equivalence class E i consists of a single element, a representation , where a ranges over the contents of all boxes in the partition λ i , and b ranges over the contents of all boxes in the partition λ j .
Remark 2.25.More generally, in the notation of [Ram03a], the partition of {Q 1 , . . ., Q ℓ } into different equivalence classes induces a partition on a weight of a calibrated representation into pages.Different pages do not interact, so a representation M i of H q,Q 1 ,...,Q ℓ (n) is unitary if and only if each M i is a unitary for every i and ℜ(a i /a j ) ≤ ℜ(q), whenever a i , a j are components of the weight a corresponding to different pages.
Since checking the condition on Remark 2.25 quickly becomes unwieldy when there are many pages, we will restrict our attention to cyclotomic Hecke algebras of the form stated in Lemma 2.23.We remark, however, that thanks to the decomposition (2.24) many of the results we prove for unitary representations, including the construction of BGG resolutions, can be extended to the general setting.For a fixed parameter q, Q 1 , . . ., Q ℓ of the cyclotomic Hecke algebra we will find all the ℓ-partitions labeling unitary representations.We define this set in Section 3 and prove that it indeed labels the unitary representations in Section 4.
3. The set of multipartitions Cali s (ℓ) 3.1.The right border multiset of an ℓ-partition with respect to an ℓ-charge.We define the right border multiset B s (λ) of λ with respect to the charge s to be the collection of integers co s (b) for each b the last box of a row of λ, with multiplicities.As in writing partitions, we will record right border multisets using exponential notation, i.e., if λ has m rows whose last boxes b satisfy co s (b) = a then we will write B s (λ) = {. . ., a m , . ..}.
If ℓ = 1 so that λ is a single partition λ and s ∈ Z, then B s (λ) is always multiplicity-free.However, for ℓ-partitions with ℓ > 1 this need not be the case as the next example shows.
3.2.The set Cali s (ℓ).We now define a set of multipartitions that, as we will see below in Theorem 4.5, provides a combinatorial description of the calibrated irreducible representations of an appropriate cyclotomic Hecke algebra.Definition 3.3.Fix e ≥ 2. Let s ∈ Z ℓ be a cylindrical charge.Define Cali s (ℓ) to be the set of all ℓ-partitions λ satisfying the following conditions: (1) (a) B s (λ) ⊂ [z, z + e − 1] for some z ∈ Z (in which case we say that it has period at most e); (b) The reading word of B s (λ) is increasing; (2) λ is cylindrical.
It is immediate that λ ∈ Cali s (ℓ) implies that λ is FLOTW, see Definition 1.15.
In the case that λ satisfies Definition 3.3(1), it is easy to check whether λ is cylindrical.Given λ = (λ 1 , λ 2 , . . ., λ ℓ ), if λ j = ∅ then define b j min to be the box of smallest content in λ j .That is, b j min is the leftmost box of the bottom row of λ j .For each j = 1, . . ., ℓ such that λ j = ∅, let h j be the number of nonzero rows of λ j .For λ j = ∅, we have co s (b j min ) = −h j + 1 + s j .
Proof.First we show that any λ satisfying (1) and ( 2) is cylindrical.We have that (1) holds if and For the converse, suppose that λ is cylindrical.Suppose j ∈ {1, . . ., ℓ − 1} such that λ j+1 = ∅.To verify condition (1), we need to show that h j+1 ≤ s j+1 − s j .Suppose not, then there exists k ≥ 1 such that h j+1 = s j+1 − s j + k.Definition 1.15(2)(a) tells us that Since B s (λ) is increasing, the charged content of the rightmost box of the bottom row of λ j+1 is greater than the charged content of the rightmost box of the top row of λ j .This yields the inequality: h j+1 , a contradiction.Next, we verify condition (2) by a similar argument.If λ 1 = ∅ then using that s is cylindrical together with condition (1) that was just proved, we have s ℓ < s 1 + e ≤ s j−1 + e < co s (b j min ) + e, where j ≥ 2 is minimal such that λ j = ∅.So we can assume j = 1 i.e. λ 1 = ∅.We want to show that s ℓ < co s (b 1 min ).We have co s (b 1 min ) = −h 1 + 1 + s 1 + e, so we need to show that h 1 ≤ s 1 − s ℓ + e. Suppose not, then we can write h 1 = s 1 − s ℓ + e + k for some k ≥ 1.By the assumption that λ is FLOTW, we have On the other hand, by the assumption that λ satisfies Definition 3.3(1), it holds that the largest element of B s (λ) is less than the smallest element of B s (λ) plus e. Thus λ ℓ h 1 , a contradiction with λ ℓ 1 ≥ λ 1 h 1 that followed above from λ being FLOTW.This concludes the proof.
3.3.Multiplicity-free right border sets and semi-infinite Young diagrams.Let I ⊂ Z be a finite subset of the integers.Write I = {i h < i h−1 < . . .< i 2 < i 1 }.We define the semi-infinite Young diagram with right border set I to be the set of boxes b = (x, y), 1 ≤ x ≤ h and y ≤ i x + x, and we denote it by Y (I).We will think of x as the row and y as the column, and will draw the rows as descending as in our convention for ordinary Young diagrams Fix I = {i 1 , . . ., i h } ⊂ Z and some e ∈ N such that e > i 1 − i h and e > h.In Example 3.7, we may take any e ≥ 7.For any ℓ ≥ 1, we will construct a particular set of charged ℓ-partitions λ with cylindrical charge s and such that B s (λ) = I.First, we choose s 1 ∈ Z such that s 1 ≤ i h .Then, we choose a box b 1 = (x 1 , y 1 ) ∈ Y (I) such that co I (b) = s 1 .Thus b 1 lies on or to the left of the northwest-southeast diagonal ending at the bottom right box of the diagram Y (I).The box b 1 will be the top left corner of λ 1 .We then take λ 1 to be the partition consisting of all boxes in Y (I) below and to the right of b 1 .That is: We then mark the boxes in Y (I) of content α, where α = co s 1 (b 1 min ).Here, b 1 min is the bottomleftmost box of λ 1 , its box of smallest content.A choice of s 1 and b 1 is illustrated schematically in Figure 4 below, with i h and α also indicated.If b 1 is in the top row of Y (I), then we stop here: we have constructed a charged 1-partition with right border I as above.Otherwise, we go to the next step.We choose some s 2 such that s 1 < s 2 < α + e. Observe that since Y (I) has less than e rows, this is always possible.On the diagonal of boxes with content s 2 , we pick a box b 2 = (x 2 , y 2 ) such that co(b 2 ) = s 2 , x 2 < x 1 , and y 2 ≥ y 1 .If s 2 = α + e − 1 then we require that x 2 = 1, that is, that b 2 is in the top row.We then take the partition λ 2 to consist of those boxes of Y (I) lying below and right of b 2 and strictly above λ 1 : If x 2 = 1, that is, if b 2 is not in the top row of the diagram, then we continue the process with Step 3 choosing s 3 and λ 3 ... At Step i of this process, which occurs if b i−1 is not in the top row of Y (I), we choose s i such that s i−1 < s i < α + e, and we choose b i = (x i , y i ) ∈ Y (I) satisfying co(b i ) = s i , x i < x i−1 , and y i ≥ y i−1 .We require that x i = 1 if s i = α + e − 1.We then define λ i as The process terminates after ℓ ≤ h steps since Y (I) has h rows and each step chooses a box b i in a row above the row containing b i−1 .We take λ = (λ 1 , λ 2 , . . ., λ ℓ ) and s = (s 1 , s 2 , . . ., s ℓ ).Then s satisfies s 1 < s 2 < . . .< s ℓ < α + e ≤ s 1 + e so is cylindric, and each λ j is non-empty, 1 ≤ j ≤ ℓ.
For λ ∈ Cali s (ℓ), removing all empty components λ j = ∅ yields a charged m-partition obtained from Y (I) by the procedure of Section 3.3 with I = B s (λ) and where m = ℓ − #{λ j = ∅}.Conversely, for a given I ⊂ Z such that I ⊂ [z, z + e − 1] for some z and |I| < e, we can start from some λ ∈ Cali s (ℓ) produced from Y (I), then insert empty components to obtain some µ ∈ Cali s ′ (ℓ ′ ), ℓ ′ > ℓ, with cylindrical charge s ′ ∈ Z ℓ ′ such that {s 1 , . . ., s ℓ } ⊂ {s ′ 1 , . . ., s ′ ℓ ′ }.The components of the charge for the empty components must be chosen so that the ℓ ′ -partition remains cylindrical, i.e. by respecting the conditions in Lemma 3.4.
3.4.Skew shapes.From now on, when we talk about the Young diagram of λ ∈ Cali s (ℓ), we will mean the embedding of the Young diagram of λ in Y (B s (λ)) given by stacking the λ j 's so that their right borders make up the right border of Y (B s (λ)).We now jettison the semi-infinite blank region to the left of the λ j 's to obtain a diagram as in the leftmost depicted in Figure 7.
Convention 3.9.We will write Y s (λ) for the Young diagram of λ ∈ Cali s (ℓ) stacked as in the leftmost diagram in Figure 7.
Suppose λ ∈ Cali s (ℓ).If we "forget components" in its Young diagram Y s (λ), we get a skew partition.In the picture of the Young diagram of the charged 3-partition above, simply forget the Suppose that every component λ j of λ is nonempty.Then we may interpret Definition 3.3 as saying that when we express Y s (λ) as µ \ ν for some partitions µ and ν, then µ and ν both belong to Cali(1).By [Ruf06], µ ∈ Cali(1) if and only if the irreducible representation S µ of the finite Hecke algebra H q (S n ) for q = exp(2πi/e) is calibrated.By [Ram03a], the calibrated representations of the affine Hecke algebra H q (S n ) for generic values of q are labeled by skew Young diagrams µ \ ν.The condition for an ℓ-partition to be in λ ∈ Cali s (ℓ) thus arises as the intersection of two combinatorial conditions: on the one hand, the skew shape condition identifying calibrated representations of the affine type A Hecke algebra for q not a root of unity; and on the other hand, the condition identifying calibrated representations of the finite type A Hecke algebra for q = exp(2πi/e), applied to the left and right borders of the skew shape.Now, if we further allow λ to contain empty components λ j = ∅, then we just need to fine-tune this description by the conditions on the charges of these empty components given by Definition 1.15 and Lemma 3.4.These extra conditions on where the empty components may occur and with what charge make sure that the resulting charge s and ℓ-partition λ remain cylindrical.
Remark 3.10.We remark that a single skew diagram can often be broken into many different charged multipartitions in an intuitive fashion.This is illustrated in Figure 8.
To finish this section, we relate our constructions to the crystal operators ẽi of Section 1.2.Lemma 3.11.Fix e ≥ 2 and a cylindrical charge s ∈ Z ℓ .The sl e -crystal operators ẽi , i ∈ Z/eZ, preserve the set of ℓ-partitions satisfying Definition 3.3(1).
Proof.Let λ be an ℓ-partition and s ∈ Z ℓ such that λ satisfies Definition 3.3(1).Then B s (λ) contains at most one element of residue i for each i ∈ Z/eZ.The set {co s (b) | b is a removable box of λ} is a subset of B s (λ).Thus λ has at most one removable i-box for each i ∈ Z/eZ.Suppose i ∈ Z/eZ such that ẽi (λ) = 0. Then λ has exactly one removable i-box, call it b.We have co s (b) ∈ B s (λ).
We claim that co s (b)−1 / ∈ B s (λ).Observe that B s (λ) = ℓ j=1 B s j (λ j ).Let us check that co s (b)−1 / ∈ B s j (λ j ) for each j = 1, . . ., ℓ.Let j(b) ∈ {1, . . ., ℓ} be the integer such that b ∈ λ j(b) , i.e. j(b) is the component of λ containing the box b.Since b is a removable box of λ j(b) , co s (b) − 1 / ∈ B s j(b) (λ j ).If j(b) < j < ℓ, then x ∈ B s j (λ j ) implies x > co s (b) by Definition 3.3(1)(b), so co s (b) − 1 / ∈ B s j (λ j ) for all j > j(b).Suppose there exists j < j(b) such that co s (b) − 1 ∈ B s j (λ j ).Then co s (b) − 1 must be the maximal element of B s j (λ j ) and co s (b) the minimal element of B s j(b) (λ j(b) ) ( and for any j < k < j(b) we must have λ k = ∅) in order that Definition 3.3(1)(b) be satisfied.Thus co s (b) − 1 is the charged content of the last box in the top row of λ j .The top row of any partition always has an addable box.Therefore λ j has an addable box of charged content co s (b) ∼ = i mod e.Since b is the only removable i-box in λ and j(b) > j, the i-word of λ contains the subword −+.It follows that b is not good removable, contradicting the assumption that ẽi (λ) = 0.

Calibrated representations of cyclotomic Hecke algebras at roots of unity
4.1.Irreducible representations of cyclotomic Hecke algebras at roots of unity.Recall that the cyclotomic Hecke algebra (or Ariki-Koike algebra) is the following quotient of the affine Hecke algebra: .
Fix e ≥ 2, ℓ ≥ 1, and s ∈ Z ℓ .Set q = exp(2πi/e) and Q i = q s i .For such a choice of parameters (sometimes called "integral parameters" in the literature), we will denote H q,Q 1 ,...,Q ℓ (n) by H e,s (n).
In the case that s is a cylindrical charge (see Definition 1.1), Foda, Leclerc, Okado, Thibon, and Welsh gave a closed-form description of these ℓ-partitions.Recall the definition of the set of FLOTW ℓ-partitions (Definition 1.15).
Theorem 4.2.[FLO+99] Let e ≥ 2 and let s ∈ Z ℓ be a cylindrical charge.Then the irreducible representations of H e,s (n) are labeled by the FLOTW ℓ-partitions of size n.
The goal of this section is to give the analogous closed-form description of the ℓ-partitions labeling the irreducible calibrated representations of H e,s (n) by the set of ℓ-partitions Cali s (ℓ) (see Definition 3.3).First, we will need the following lemma, which uses Theorem 4.2.
The analogue of Theorem 4.1 for the irreducible calibrated representations of H e,s (n) identifies the ℓ-partitions labeling them in terms of certain paths in the sl e -crystal.
Theorem 4.4.[Gro99] Let e ≥ 2 and let s ∈ Z ℓ .The irreducible calibrated representations of H e,s (n) are labeled by the ℓ-partitions of the form λ = fin fi n−1 . . .fi 2 fi 1 ∅ such that i = i + 1 for all i = i 1 , . . ., i n−1 in any such expression for λ.
That is, we consider all possible ways to build up a FLOTW ℓ-partition λ from the empty ℓpartition ∅ by adding one box at a time such that at each step, the box we add is the good addable box for its residue.The theorem says that if λ labels an irreducible calibrated representation, then in all such sequences building up λ one box at a time, we never add an i-box immediately followed by another i-box.
We now arrive at the main result of this section, which is the analog of Theorem 4.2 for calibrated representations.
Theorem 4.5.Let e ≥ 2 and let s ∈ Z ℓ be a cylindrical charge.Then the irreducible calibrated representations of H e,s (n) are labeled by the ℓ-partitions in Cali s (ℓ) of size n.
Proof.We will prove the statement by induction on n.The base case is n = 1.By Theorem 4.4, if |λ| = 1 then λ labels an irreducible calibrated representation if and only if λ is FLOTW.By Definition 1.15 and Lemma 3.4, if |λ| = 1 then λ ∈ Cali s (ℓ) if and only if λ is FLOTW.Now suppose by induction that for all FLOTW ℓ-partitions λ of n, it holds that λ labels an irreducible calibrated representation of H e,s (n) if and only if λ ∈ Cali s (ℓ).
The remainder of the proof is dedicated to showing that if λ is a FLOTW ℓ-partition of size n + 1 that labels an irreducible calibrated representation of H e,s (n + 1), then λ ∈ Cali s (ℓ).Suppose that µ is FLOTW, |µ| = n + 1, and µ labels an irreducible calibrated representation.By Theorem 4.4, for every expression µ = fi n+1 fin . . .fi 2 fi 1 ∅ it holds that i k+1 = i k for all k = 1, . . ., n.We need to show that µ satisfies Definition 3.3(1).By induction, µ = fi λ for some λ ∈ Cali s (ℓ) and some i ∈ Z/eZ.Either fi adds a box to a non-zero row of λ, or fi creates a new row.In the case that fi adds a box to an already existing row, the result is forced by the induction hypothesis that λ ∈ Cali s (ℓ) using arguments similar to those in the proof of Lemma 3.11.This is straightforward to check.The work consists in dealing with the case that fi adds the good i-box in a new row of λ.Thus we suppose from now on that fi adds a new row to λ, i.e. that |B s (µ)| = |B s (λ)| + 1.We will show that if Definition 3.3(1)(a) or Definition 3.3(1)(b) fails for fi λ = µ, then there exists an expression fi n+1 fin . . .fi 2 fi 1 ∅ = µ with i k = i = i k+1 for some k ∈ {1, . . ., n}.
First, suppose Definition 3.3(1)(a) fails, so suppose B s (µ) ⊂ [z, z + e − 1] for all z ∈ Z.Let b be the i-box added by fi to λ, i.e. µ = fi λ ∪ {b}, and let j(b) ∈ {1, . . ., ℓ} be such that b is a box in µ j(b) .First, we show that co s (b) must be the smallest element of B s (µ).Suppose that co s (b) is neither the smallest nor the largest element of B s (µ).Since B s (λ) ⊂ [z, z + e − 1] for some z ∈ Z, this implies that B s (µ) = [z, z + e − 1] for some z ∈ Z, and thus co s (b) − 1 ∈ B s (λ).Then co s (b) − 1 is the content of the box of largest content in µ k for some k < j(b).But then µ k has an addable box of content co s (b).It follows that the good addable box of λ is then in µ k , not in µ j(b) , contradicting the assumption.Next, since λ ∈ Cali s (ℓ) then by Lemma 3.4 we have s j(b) − co s (b min ) < e where b min is the box of smallest charged content in λ.Thus if co s (b) is the largest element of B s (µ) then B s (µ) ⊆ [z, z + e − 1] and we repeat the argument above to get a contradiction.We conclude that co s (b) the smallest element of B s (µ), and moreover (again by the same argument), co s (b) = co s (b max ) − e + 1.
It follows that there exists a unique w ∈ Z such that w = co s (b) + qe for some positive integer q, w − e < x < w + e for all x ∈ B s (λ) = B s (µ) \ {co s (b)}, and there is some x ∈ B s (λ) with x ≥ w.We observe that λ cannot have an addable box of content w in one of its nonzero rows, for if it did then ẽi would have added this box rather than b.Thus if w − 1 ∈ B s (µ), it must also hold that w ∈ B s (µ).
Step 1. Case (i).Suppose w − 1 / ∈ B s (µ).Consider B s (µ) ∩ Z ≥w , and the corresponding boxes in the right border of µ that are of charged content at least w.Let r = |B s (µ) ∩ Z ≥w |.Since the reading word of B s (λ) is increasing, these boxes belong to the top r rows of Y s (λ).We have 0 < r ≤ e − 1.If B s (µ) ∩ Z ≥w = [w, w + r − 1], proceed to Step 2. Otherwise, let y be the minimal element of B s (µ) ∩ Z ≥w .Apply ẽ := ẽi+1 ẽi+2 . . .ẽy−1 ẽy to µ.This removes a horizontal strip of y − w + 1 boxes from the r'th row of Y s (µ), because µ = λ ∪ {b} has no addable boxes of these residues with charged content w or greater nor does it have any (other) removable boxes of these residues.Applying ẽi ẽi to ẽµ now removes b and the removable i-box in row r yielding ν such that µ = fy fy−1 . . .fi+2 fi+1 fi fi ν, a contradiction.This step of the proof is illustrated in Figure 9.
Step 2. Recall that w ∼ = co s (b) ∼ = i mod e.The ℓ-partition ν has no addable i-box in any of its nonzero rows.It has two removable i-boxes: b, and the box of content w in its border.Since fi added the box co s (b) to λ, ν has no addable i-box bigger than b in an empty row.The i-word for ν is thus some number of plusses followed by −−.It follows that ẽi ẽi removes the boxes of charged contents w and co s (b) from ν.This shows that µ = fi fi (ẽ i ẽi ν), a contradiction with the assumption that µ = fi λ labels a calibrated representation.Therefore Definition 3.3(1)(a) must hold if fi λ satisfies the condition of Theorem 4.4.
Finally, we suppose that fi λ = 0 satisfies the condition of Theorem 4.4 but that Definition 3.3(1)(b) fails for fi λ.The argument by contradiction is similar to the one checking Definition 3.3(1)(a).Define b and j(b) be as above.By assumption the reading word of B s (µ) is not increasing, but the reading word of B s (λ) is.By the previous step of the proof, we know that the elements of B s (µ) all belong to an interval [z, z + e − 1] for some z ∈ Z. Since we are in the case that adding b creates a new row in component j(b), and h(λ) is s-admissible by Lemma 3.5 we have co s (b) < co s (b ′ ) for all co s (b ′ ) ∈ B s k (µ k ), k ≥ j(b).Thus, there must exist a box b ′ ∈ µ m = λ m for some m < j(b) such that b ′ is the last box in its row and co s (b) ≤ co s (b ′ ), and we take m to be minimal such that this happens.Now we remove either a horizontal strip, a vertical strip, or a combination of such from λ m by crystal operators in order to arrive at a situation where ẽi can be applied twice in a row to get a non-zero ℓ-partition, as in previous part of the proof.We arrive at that situation in the following ways depending on how the diagonal of charged content co s (b) intersects the Young diagram of λ m in Y s (λ).
First, if co s (b) ≤ co s (b ′ ) for all boxes b ′ the last in their rows in λ m then we remove all boxes of content larger than co s (b) from the bottom row of λ m .We then have two removable boxes of residue i.These are successively good removable unless the largest element of B s (λ) has residue i − 1.If the latter is the case then we remove the topmost vertical strip in the border of Y s (λ).Then we may remove the two boxes of residue i by applying ẽi twice in a row.
Otherwise, i is the residue of a box in the right border ribbon of λ m .We consider whether it occurs in an arm, a leg, or a corner of this ribbon.We then remove, respectively, the arm to its right, the leg below it, or the arm and the leg below it and to its right.In case the bottom box of the leg below it is the minimal element z of B s (λ) and both z, z + e − 1 ∈ B s (λ) then again we have to remove the topmost vertical strip in the border of Y s (λ) before we can remove that leg via crystal operators ẽj .We then arrive again at the situation that we may apply ẽi twice in a row to remove two boxes of residue i.Therefore, the irreducible representation labeled by µ is not calibrated.

Alcove geometries and path combinatorics
We now set about providing a homological construction of the calibrated representations via BGG resolutions.This means understanding these simple modules in terms of the Specht theory of the cyclotomic Hecke algebra.The homological and representation theoretic structure of the cyclotomic Hecke algebra is governed by strong uni-triangularity properties -thus we can understand a given calibrated simple D s (λ) in terms of the Serre subcategory arising from the poset Λ = {µ | µ λ} ⊆ P ℓ (n).We will cast each µ ∈ Λ as a point in an h-dimensional alcove geometry under the action of the affine symmetric group S h ; the calibrated simple D s (λ) will belong to the fundamental alcove under this action.We regard S h as a Coxeter group, we let ℓ : S h → N denote the corresponding length function and we let ≤ denote the strong Bruhat order on S h .

The alcove geometry. Set
)+i denote a formal symbol, and define an h-dimensional real vector space Rε i,m and E h to be the quotient of this space by the one-dimensional subspace spanned by We have an inner product , on E h given by extending linearly the relations ε i,p , ε j,q = δ i,j δ p,q for all 1 ≤ p, q ≤ ℓ, 1 ≤ i ≤ h p and 1 ≤ j ≤ h q , where δ i,j is the Kronecker delta.We identify λ ∈ P h (n) with an element of the integer lattice inside E h via the map We let Φ denote the root system of type A h−1 consisting of the roots and Φ 0 denote the root system of type We choose ∆ (respectively ∆ 0 ) to be the set of simple roots inside Φ (respectively Φ 0 ) of the form ε t − ε t+1 for some t.Given r ∈ Z and α ∈ Φ we define s α,re to be the reflection which acts on E h by s α,re x = x − ( x, α − re)α The group generated by the s α,0 with α ∈ Φ (respectively α ∈ Φ 0 ) is isomorphic to the symmetric group S h (respectively to S h := S h 1 × • • • × S h ℓ ), while the group generated by the s α,re with α ∈ Φ and r ∈ Z is isomorphic to S h , the affine Weyl group of type A h−1 .We set α 0 = ε h − ε 1 and Π = ∆ ∪ {α 0 }.The elements S = {s α,0 : α ∈ ∆} ∪ {s α 0 ,−e } generate S h .
Notation 5.1.We shall frequently find it convenient to refer to the generators in S in terms of the elements of Π, and will abuse notation in two different ways.First, we will write s α for s α,0 when α ∈ ∆ and s α 0 for s α 0 ,−e .This is unambiguous except in the case of the affine reflection s α 0 ,−e , where this notation has previously been used for the element s α,0 .As the element s α 0 ,0 will not be referred to hereafter this should not cause confusion.Second, we will write α = ε i − ε i+1 in all cases; if i = h then all occurrences of i + 1 should be interpreted modulo h to refer to the index 1.
We shall consider a shifted action of the affine Weyl group S h on E h,l by the element that is, given an element w ∈ S h , we set w • x = w(x + ρ) − ρ.This shifted action induces a well-defined action on E h ; we will define various geometric objects in E h in terms of this action, and denote the corresponding objects in the quotient with a bar without further comment.We let E(α, re) denote the affine hyperplane consisting of the points Note that our assumption that e > h 1 + • • • + h ℓ implies that the origin does not lie on any hyperplane.Given a hyperplane E(α, re) we remove the hyperplane from E h to obtain two distinct subsets E > (α, re) and E < (α, re) where the origin lies in E < (α, re).The connected components of are called chambers.The dominant chamber, denoted E + h , is defined to be The connected components of are called alcoves, and any such alcove is a fundamental domain for the action of the group S h on the set Alc of all such alcoves.We define the fundamental alcove, which we denote by F h ⊆ E h , to be the alcove containing the origin (which is inside the dominant chamber) and we set We have a bijection from S h to Alc given by w −→ wF h .Under this identification Alc inherits a right action from the right action of S h on itself.Consider the subgroup The dominant chamber is a fundamental domain for the action of S h on the set of chambers in E h .We let S h denote the set of minimal length representatives for right cosets S h \ S h .So multiplication gives a bijection S h × S h → S h .This induces a bijection between right cosets and the alcoves in our dominant chamber.
If the intersection of a hyperplane E(α, re) with the closure of an alcove A is generically of codimension one in E h then we call this intersection a wall of A. The fundamental alcove F h has walls corresponding to E(α, 0) with α ∈ ∆ together with an affine wall E(α 0 , −e).We will usually just write E(α) for the walls E(α, 0) (when α ∈ ∆) and E(α, −e) (when α = α 0 ).We regard each of these walls as being labelled by a distinct colour (and assign the same colour to the corresponding element of S).Under the action of S h each wall of a given alcove A is in the orbit of a unique wall of F h , and thus inherits a colour from that wall.We will sometimes use the right action of S h on Alc.Given an alcove A and an element s ∈ S, the alcove As is obtained by reflecting A in the wall of A with colour corresponding to the colour of s.With this observation it is now easy to see that if w = s 1 . . .s t where the s i are in S then wF h is the alcove obtained from F h by successively reflecting through the walls corresponding to s 1 up to s t .5.2.Paths in the geometry.We now develop a path combinatorics inside our geometry.Given a map p : {1, . . ., n} → {1, . . ., h} we define points P(k) ∈ E h by for 1 ≤ i ≤ n.We define the associated path by P = (∅ = P(0), P(1), P(2), . . ., P(n)) and we say that the path has shape π = P(n) ∈ E h .We also denote this path by Given λ ∈ E h , we let Path(λ) denote the set of paths of length n with shape λ.We define Path h (λ) to be the subset of Path(λ) consisting of those paths lying entirely inside the dominant chamber; i.e. those P such that Given paths P = (ε p(1) , . . ., ε p(n) ) and Q = (ε q(1) , . . ., ε q(n) ) we say that P ∼ Q if there exists an α ∈ Φ and r ∈ Z and s ≤ n such that In other words the paths P and Q agree up to some point P(s) = Q(s) which lies on E(α, re), after which each Q(t) is obtained from P(t) by reflection in E(α, re).We extend ∼ by transitivity to give an equivalence relation on paths, and say that two paths in the same equivalence class are related by a series of wall reflections of paths and given S ∈ Path h (n) we set [S] = {T ∈ Path h (n) | S ∼ T}.Given a path P we define res(P) = (res P (1), . . ., res P (n)) where res P (i) denotes the residue of the box labelled by i in the tableau corresponding to P. We have that res(P) = res(Q) is and only if P ∼ Q.
Definition 5.2.Given a path S = (S(0), S(1), S(2), . . ., S(n)) we set deg s (S(0)) = 0 and define where d(S(k), S(k − 1)) is defined as follows.For α ∈ Φ we set d α (S(k), S(k − 1)) to be Definition 5.3.Given two paths we define the naive concatenated path For λ ∈ P h (n), we identify Path h (λ) with the set of standard λ-tableaux in the obvious manner (see [BCH20] for more details).This identification preserves the grading.This identification is best illustrated via an example: Example 5.4.The tableau T 1,λ in Example 1.9 corresponds to the path 5.3.The Bott-Samelson truncation and Soergel diagrammatics.We now recall the construction an idempotent subalgebra of H h (n, s) which is isomorphic to Elias-Williamson's diagrammatic Hecke categories.In order to do this, we must restrict our attention to paths labelled by (enhanced) words in the affine Weyl group.Definition 5.5.We will associate alcove paths to certain words in the alphabet where s ∅ = 1.That is, we will consider words in the generators of the affine Weyl group, but enriched with explicit occurrences of the identity in these expressions.We refer to the number of elements in such an expression (including the occurrences of the identity) as the enhanced length of this expression.We say that an enriched word is reduced if, upon forgetting occurrences of the identity in the expression, the resulting word is reduced.
Given a path P between points in the principal linkage class, the end point lies in the interior of an alcove of the form wF h for some w ∈ S h .If we write w as a word in our alphabet, and then replace each element s α by the corresponding non-affine reflection s α in S h to form the element w ∈ S h then the basis vectors ε i are permuted by the corresponding action of w to give ε w(i) , and there is an isomorphism from E h,l to itself which maps F h to wF h such that 0 maps to w • 0, coloured walls map to walls of the same colour, and each basis element ε i map to ε w(i) .Under this map we can transform a path Q starting at the origin to a path starting at w • 0 which passes through the same sequence of coloured walls as Q does.
We now define the building blocks from which all of our distinguished paths will be constructed.We begin by defining certain integers that describe the position of the origin in our fundamental alcove.
Definition 5.7.Given α ∈ Π we define b α to be the distance from the origin to the wall corresponding to α, and let b ∅ = 1.Given our earlier conventions this corresponds to setting We let δ k = ((k h 1 ), (k h 2 ), . . ., (k h ℓ )) and we note that these multipartitions always lie in the principal linkage class.We sometimes write δ α for the element δ bα .We can now define our basic building blocks for paths.
Definition 5.8.Given α = ε i −ε i+1 ∈ Π, we consider the multicomposition s α •δ α with all columns of length b α , with the exception of the ith and (i + 1)st columns, which are of length 0 and 2b α , respectively.We set where .denotes omission of a coordinate.Then our distinguished path corresponding to s α is given by The distinguished path corresponding to ∅ is labelled by P ∅ ∈ Path(δ) and is fixed to be any choice of tableau T m,δ for which 1 ≤ m ≤ ℓ is a step change.We set P ∅ = (P ∅ ) bα .Remark 5.9.If ℓ is a step change, then we can take P ∅ = (ε 1 , ε 2 , . . ., ε h ) and indeed this is the path used in [BCH20] (where it is implicitly assumed that ℓ is a step change).We further remark that one can always reorder the charge s ∈ Z ℓ to obtain some s ∈ Z ℓ for which ℓ is a step change (using the trivial algebra isomorphism H n (s) ∼ = H n ( s)).
We are now ready to define our distinguished paths for general words in our alphabet.Definition 5.10.We now define a distinguished path P w for each word w in our alphabet S ∪ {1} by induction on the enhanced length of w.If w is s ∅ or a simple reflection s α we have already defined the distinguished path in Definition 5.8.Otherwise if w = s α w ′ then we define If the enriched word w is reduced, then the corresponding path P w is said to be a reduced path.Definition 5.11.Given α ∈ Π we set the path obtained by reflecting the second part of P α in the wall through which it passes.
We let Std n,s (λ) ⊆ Std h (λ), to be the set of all alcove-tableaux which can be obtained by contextualised concatenation of paths from the set Figure 10.The first two diagrams are a path P α walking through an α-hyperplane, and a path P ♭ α obtained by reflecting P α through this α-hyperplane.The final diagram is the path P ∅ .

Categorification and BGG resolutions
We now introduce the graded diagrammatic algebras which provide the necessary context for constructing our BGG resolutions.
6.1.1.The quotient algebras of interest.We set e T := e res(T) ∈ H n (s).For λ ∈ P ℓ n , we set We remark that y λ = e T (m,λ) for λ ∈ P h (n).Given h ∈ N ℓ we define (6.5) Given S, T ∈ Std(λ) and w any fixed reduced word for w S T we let ψ S T := e S ψ w e T .We have already seen that if λ is a calibrated ℓ-partition, then h(λ) is s-admissible.Definition 6.6.Given s ∈ Z ℓ , we let h = (h 1 , . . ., h ℓ ) ∈ N ℓ be s-admissible.We define H h (n, s) := H n (s)/H n (s)y h H n (s).(3) If S, T ∈ Path h (λ), for some λ ∈ P h (n), and a ∈ H h (n, s) then there exist scalars r SU (a), which do not depend on T, such that where the scalars r SU (a) are the scalars appearing in (3) of Theorem 6.7.
Remark 6.10.The modules S s (λ) are obtained (via specialisation) from the usual semisimple modules over C[q, Q 1 , . . ., Q ℓ ] with the same multipartition labels.We remark that the integral form (in the modular system by which we specialise) is constructed from the cylindric charge in [Bow17,BCHM20] and can be seen as coming the quiver Cherednik algebra associated to s ∈ Z ℓ .
Proof.Any λ ∈ Cali s (ℓ) must satisfy that h(λ) = h is s-admissible (by Corollary 3.5) and so it is enough to restrict our attention to λ ∈ E h for some s-admissible h ∈ N ℓ .For λ ∈ E h the condition that the border strip is increasing is equivalent to the condition that λ ∈ E < (ε i − ε i+1 , e) for 1 ≤ i < ℓ.Similarly, the condition that λ has period at most e is equivalent to the condition that λ ∈ E < (ε 1 − ε h , −e).The result follows.
Definition 6.12.Given α ∈ Π we define the corresponding Soergel idempotent, 1 α , to be a frame of width 1 unit, containing a single vertical strand coloured with α ∈ Π.We define 1 ∅ to be an empty frame of width 1 unit.For w = s α (1) . . .s α (p) with α (i) ∈ S ∪ {1} for 1 ≤ i ≤ p, we set to be the diagram obtained by horizontal concatenation.which we often denote these diagrams by respectively along with their flips through the horizontal axis and their isotypic deformations such that the north and south edges of the graph are given by the idempotents 1 w and 1 w ′ respectively.
Here the vertical concatenation of a (w, w ′ )-Soergel diagram on top of a (v, v ′ )-Soergel diagram is zero if v = w ′ .We define the degree of these generators (and their flips) to be 0, 1, −1, 0, and 0 respectively.We let * denote the map which flips a diagram through its horizontal axis.
Suppose that w and w ′ are both words with the same underlying permutation and that they can be obtained from one another by a sequence of applications of the braid relations of S h (i.e.without applying the quadratic relation); we let braid w w ′ (or braid Pw P w ′ if we wish to emphasise the corresponding paths) denote the product of the corresponding sequence of the braid generator Soergel diagrams (the final two pictures in 6.14).Definition 6.15.Let k be an integral domain.We define the diagrammatic Bott-Samelson endomorphism algebra, S h (n, s), to be the locally-unital associative k-algebra spanned by all (x, x ′ )-Soergel diagrams for x, x ′ ∈ Λ h (n, s) with multiplication given by vertical concatenation of diagrams modulo the following local relations and their duals.We have the idempotent relations, For each α ∈ S we have monochrome relations For m(α, β) = 3 and m(β, γ) = 2 we have the two-colour barbell relations and the fork-braid relations and the Jones-Wenzl relations For all diagrams D 1 , D 2 , D 3 , D 4 and all enhanced words x, y, we require the bifunctoriality relation and the monoidal unit relation Finally, we require the (non-local) cyclotomic relation spot ∅ α spot α ∅ ⊗ 1 w = 0 for all w ∈ exp(w), w ∈ S h , and all s α ∈ S h .6.2.1.Cellularity and quasi-hereditary structure.We can extend an alcove-tableau Q ′ ∈ Std n,s (λ) to obtain a new alcove-tableau Q in one of three possible ways for some α ∈ Π.The first two cases each subdivide into a further two cases based on whether α is an upper or lower wall of the alcove containing λ.
Definition 6.16.Suppose that λ belongs to an alcove which has a hyperplane labelled by α as an upper alcove wall.Let Now suppose that λ belongs to an alcove which has a hyperplane labelled by α as a lower alcove wall.Thus we can choose Theorem 6.17 ([EW16, Section 6.4]).For each λ ∈ P h (n, s), we fix an arbitrary reduced path λ) r SU (a)c U , where the scalars r SU (a) are the scalars appearing in (3) of Theorem 6.17.Theorem 6.19.For k a field, the algebra S h (n, s) is quasi-hereditary with simple modules L(λ) = ∆(λ)/rad(∆(λ)) for λ ∈ P h (n, s).Remark 6.20.We remark that the algebra S h (n, s) together with its basis given in Theorem 6.17 is an example of a based quasi-hereditary algebra with involution * , see [BS21].
6.3.The isomorphism and BGG resolutions.We are now ready to restrict our attention to regular blocks of H h (n, s).Given α a simple reflection or α = ∅, we have an associated path P α , a trivial bijection w Pα Pα = 1 ∈ S bαh , and an idempotent element of the quiver Hecke algebra and we define (6.21) Theorem 6.22 ([BCH20, Theorems A and B]).Let s ∈ Z ℓ and k be an arbitrary integral domain.Let e > h and suppose that h ∈ N ℓ is s-admissible.We have an isomorphism of graded k-algebras, Moreover, the isomorphism preserves the graded cellular structures of these algebras, that is, We now recall the construction of the BGG resolutions for certain S h (n, s)-modules.In what follows, we let λ For the remainder of the paper we label S h (n, s)-modules by the cosets w, x, y, z ∈ S h , rather than by the multipartitions as this is more convenient for indexing the homomorphisms in our resolutions.Definition 6.23.Given w, y ∈ S h , we say that (w, y) is a Carter-Payne pair if y ≤ w and ℓ(y) = ℓ(w) − 1.Let (w, x) and (x, z) be Carter-Payne pairs.If there exists a (necessarily unique) y ∈ S h such that w ≥ y ≥ z, then we refer to the quadruple w, x, y, z as a diamond.If no such y exists, we refer to the triple w, x, z as a strand.Theorem 6.24.We let λ ∈ F h (n)•w, µ ∈ F h (n)•x and suppose that (w, x) is a Carter-Payne pair.Pick an arbitrary w = σ 1 . . .σ ℓ and suppose that x = σ 1 . . .σ p−1 σ p σ p+1 . . .σ ℓ is the subexpression for x obtained by deleting precisely one element σ p ∈ S. We have that Hom S h (n,s) (∆(w), ∆(x)) is t 1 -dimensional and spanned by the map We define a complex of graded S h (n, s)-modules We will refer to this as the BGG complex.For w, x, y, z a diamond we have homomorphisms of given by our Carter-Payne homomorphisms of 6.25.By an easy variation on [BGG75, Lemma 10.4], it is possible to pick a sign ǫ(α, β) for each of the four Carter-Payne pairs such that for every diamond the product of the signs associated to its four arrows is equal to −1.For every strand w, x, z have homomorphisms We can now define the S h (n, s)-differential δ ℓ : ∆ ℓ → ∆ ℓ−1 for ℓ ≥ 1 to be the sum of the maps The underlying graded character is as follows, We emphasise that the identity coset labels the fundamental alcove in the alcove geometry, by definition (and so has length 0).That is, the alcove F h (n) which contains all the calibrated simple modules D s (λ) such that h(λ) = h.We set the length of µ ∈ F h (n) • x to be given by ℓ(µ) := ℓ(x).By the translation principle of [BCH20, Proposition 7.4], we immediately obtain the following: Theorem 6.28.Given h ∈ N ℓ and λ ∈ F h (n), we have an associated complex This complex is exact except in degree zero, where H 0 (C • (λ)) = D s (λ).The underlying graded character is as follows, By classical results for quasi-hereditary algebras, we have the following immediate corollary: Corollary 6.29.Let λ ∈ F h (n) and µ λ.We have that Finally, we now use our BGG resolutions to construct the characteristic-free bases of Theorem C. Definition 6.30.Given λ ∈ F h (n), we define Path F h (λ) to be the set of all paths The action of H n (s) is as follows: where S k↔k+1 is the tableau obtained from S by swapping the entries k and k + 1.
Proof.Since each µ λ lies in an alcove, |Rem r (µ)| ≤ 1 for r ∈ Z/eZ.By the branching rule [BCHM20, Proposition 1.26], each S s (µ) restricts to be a direct sum of Specht modules and therefore res where every simple on the righthand-side is labelled by some λ − r ∈ F h (n − 1).Thus the basis result follows by restriction.The action of the idempotents on this basis is obvious.The other zero-relations all follow because the product has non-zero degree (whereas the module D n (λ) is concentrated in degree 0).Finally, assume |res(S −1 (k)) − res −1 (S(k + 1))| > 1.The strands terminating at the kth and (k + 1)th positions on the northern edge either do or do not cross.In the former case, we can resolve the double crossing in ψ k c S without cost by our assumption on the residues and the result follows.The latter case is trivial.Finally, notice that S k↔k+1 ∈ Path F h (λ) under the assumption that |res(S −1 (k)) − res −1 (S(k + 1))| > 1.
Remark 6.32.The algebra H h (n, s) is a quasi-hereditary quotient of the Hecke algebra for s ∈ Z ℓ with respect to the s-cellular structure of [Bow17,BCHM20].Therefore H h (n, s) is Morita equivalent to the corresponding quotient of the Cherednik algebra with cylindric charge s ∈ Z ℓ (although this requires some chasing through results of [Bow17, BCHM20, Web17]) and thus one can lift all our results to the Cherednik algebra with cylindric charge s ∈ Z ℓ .This (Morita equivalent) setting is of interest because it allows one to deduce geometric applications of our results, following the machinery of [Gri21] and [BNS18, Section 9] (but we do not explore this here).Moreover, in this setting the Ext-groups calculated in Corollary 6.29 can also be computed using a cyclotomic version of Littlewood-Richardson coefficients, see [FGM21, Theorem 1.1].As the authors of [FGM21] observe, "for small values of ℓ and n, there is a certain tendency for the dimensions of the relevant Ext groups are always 0 or 1".It is not at all evident from their formula that a multiplicity one result should hold for a wide class of representations, such as those of full support.Our Corollary 6.29 provides evidence for their postulation (which they posit in the context of arbitrary charges s ∈ Z ℓ ) for representations with full support.
Finally, for level ℓ = 2 with a non-cylindric charge s ∈ Z ℓ , it is shown in [Nor20] that all unitary representations of the Cherednik algebra can be reduced to the level ℓ = 1 case.Thus for level 2, every unitary representation of a Cherednik algebra (regardless of the charge) admits a BGG resolution.
Lemma A.1.Let m = (m 1 , . . ., m n ) ∈ I n .Then, m is a weight of a calibrated H q (n)-module if and only if the following conditions are satisfied • for every i < j, if m i = m j , then m i +1, m i −1 ∈ {m i+1 , . . ., m j−1 } (that is, m is calibrated) • m 1 = 0.
• for every i > 1, {m i − 1, m i + 1} ∩ {m 1 , . . ., m i−1 } = ∅ We let C ⊆ I m be the set of weights satisfying the conditions of Lemma A.1.Note that if m ∈ C and s i is an admissible transposition of m (that is, m i − m i+1 = ±1) then s i m ∈ C. Thus, the set of irreducible calibrated H q (n)-modules is parametrized by C/ ∼.Now let U ⊆ C be the set of weights appearing in unitary H q (n)-modules.In general, U is only a proper subset of C. For certain values of q we do have U = C, as the following result shows.
Proposition A.2. Let e > 0 and let q = exp(2π √ −1/e).Then, U = C and an H q (n)-module is calibrated if and only if it is unitary.
Proof.By our choice of q, the only power of q whose real part is greater than that of q is q e = 1.Now let m ∈ C, and i < j such that m i = m j .Since q m i −m j = 1 we get ℜ(q m i −m j ) ≤ ℜ(q).The result now follows from Lemma 2.22 Remark A.3.If we take another primitve e-th root of unity, we may find calibrated representations which are not unitary.For example, if e is odd and q = exp((e − 1)π √ −1/e), then ℜ(q k ) ≥ ℜ(q) for every k ∈ Z, so any unitary H q (n)-module is 1-dimensional.
Calibrated modules for H q (n) have been classified in previous work [Kle96,Ruf06] (as well as being a special case of Theorem B).
Lemma A.4.If q is not a root of unity, then D(λ) is calibrated for any partition λ.If q is a primitive e-th root of unity, then D(λ) is calibrated if and only if |B s (λ)| < e.
Our goal is now to use Lemma A.4 to completely classify of unitary representations of H q (n).
A.3.Admissible tableaux.Note that for k ≤ n we have a natural inclusion H q (k) ⊆ H q (n) and thus we have an exact restriction functor Res n k : H q (n)-mod → H q (k)-mod.We will say that T ∈ Std(λ) is q-admissible if, for every k, the representation D(Shape(T↓ {1,...,k} )) of H q (k) is nonzero and a subrepresentation of Res n k (D(λ)).Equivalently, this means that D(λ) is nonzero and the box labeled by k is good removable box of Shape(T↓ {1,...,k} ) for every k.If q is not a root of unity then every tableau on λ is q-admissible, but this is not the case if q is a root of unity.
If D(λ) is calibrated, admissible tableaux correspond to weights as follows.Let T be an admissible tableau of λ.Then, m T = (− ct(T −1 (1)), . . ., − ct(T −1 (n))) is a weight of D(λ).This defines a bijection between weights of D(λ) and admissible tableaux on λ.Now let us denote by C the column-reading tableau on λ, that is the tableau obtained by placing {1, 2, . . ., λ t 1 } on the boxes in the first column, {λ t 1 + 1, . . ., λ t 1 + λ t 2 } on the boxes in the second column, and so on, see Definition 1.8.The following result will be very important in our arguments.
Lemma A.5. Assume that D(λ) is a calibrated H q (n)-module.Then, the column-reading tableau C on λ is admissible.
Proof.If q is not a root of unity, there is nothing to show.Let us assume that q is a primitive e-th root of unity, so that |B s (λ)| + 1 < e.We claim that C −1 (n) is a good removable box of λ.To see this note that, since |B s (λ)| + 1 < e, λ has at most one removable box of each residue, so the claim will follow if we check that there is no addable box of the same residue as C −1 (n) to the left of C −1 (n).But this is clear since the last column of λ has at most (e − 1)-boxes.Now, D(λ \ {C −1 (n)}) is a calibrated H q (n − 1)-module.The column-reading tableau of λ \ {C −1 (n)} is simply the restriction of the column-reading tableau of λ.So the result follows by an inductive argument.
A.4. Unitary loci.In this section, we classify the unitary representations of H q (n) for any q ∈ C × , n > 0. Recall that it only makes sense to speak about unitary representations when q lies in the unit circle, so the following definition is sensible.Sto10,Sto09]).Let λ ⊢ n be a partition.We define the unitary locus of λ to be U (λ) := {c ∈ (−1/2, 1/2] : D(λ) = 0 is a unitary representation of H exp(2π √ −1c) (n)} We will classify unitary representations via a complete, explicit description of the unitary locus of every partition.To state our result, we first fix some notation.For λ ∈ P 1 (n) with h(λ) = h ∈ N, we define the hook length of a node (i, j) ∈ λ as follows Remark A.8.We remark that Theorem A.7 has appeared in work of Stoica [Sto09], with two differences.First, our conventions are dual to those of Stoica, so the statements differ by taking the transpose of λ.Second, and most importantly, there is an oversight in [Sto09, Theorem 4.2], which does not consider elements of the form d/m in case (3) above.But in this case the representation D(λ) is 1-dimensional, a fortiori unitary.The oversight in [Sto09] seems to stem from the computation of the branching rule for the Hecke algebra in [Sto09,Proposition 4.3].Finally, we remark that Venkateswaran has computed the unitary loci under the assumption that q is not a root of unity in [Ven20].
The proof of Theorem A.7 is contained in the next several lemmas.To start, we have the following easy result, which covers cases (1) and (2).
Proof.The representation D(n) is always 1-dimensional, while the representation D(1 n ) is 1dimensional whenever it is defined, and it is defined if and only if q is not an e-th root of unity with e ≤ n.The result follows.
From now on, we will assume that λ is neither the trivial nor the sign partition.Let us start with the case when q is not a root of unity, equivalently, c is irrational.
Proof.From the column-reading tableau of λ, we can see from Lemma 2.22 that D(λ) is H q (n)unitary if and only if ℜ(q i ) ≤ ℜ(q) for every i = 1, . . ., ℓ.The result follows.
Remark A.11. Lemma A.10 also follows from the main result of [Ven20], where the signature of the form •, • on D(λ) is computed under the assumption that q is not a root of unity.Now we need to consider the case of rational c, i.e., when q is a root of unity.First, we consider the case where |c| ≤ 1/ℓ.Lemma A.12.Let λ = (1 n ), (n) and c ∈ [−1/ℓ, 1/ℓ].Then, c ∈ U (λ).
Proof.In view of Lemma A.10, we only need to consider the case where q := exp(2π √ −1c) is a root of unity.Let e > 0 be minimal such that q e = 1.Since c ∈ [−1/ℓ, 1/ℓ], we have that e ≥ ℓ.If either λ t 1 < ℓ or ℓ < e, it follows from Lemma A.4 that D λ is calibrated and then, by Lemma A.5, that the column-reading tableau of λ is e-admissible, The result now follows just as in the proof of Lemma A.10.If λ 1 = ℓ = e, then the partition is the one-row partition (e).But this is not e-restricted and we are done.

Figure 2 .
Figure 2. The 3-partitions from Examples 3.1 and 3.2 together with their contents.We have highlighted the boxes at the end of each row in the Young diagram of λ.

Figure 3 .
Figure 3.The diagram Y (I) as in Example 3.7 and its truncation at the column whose topmost box has content 0 to obtain the Young diagram of λ = (7, 6, 5 3 ).

Figure 4 .
Figure 4. Illustration of the first step of producing a calibrated multipartition from a semi-infinite Young diagram

Figure 9 .
Figure 9. Illustration of the proof of Theorem 4.5, induction step verifying that failure of Definition 3.3(1)(a) implies failure to be calibrated.Step 1, Case (i).
E k be the different equivalence classes.Thanks to work of Dipper and Mathas, see e.g.[Ari02, Theorem 13.30] we have a category equivalence

r
SU (a)ψ UT (mod H ⊲λ ), where H ⊲λ is the k-submodule of H h (n, s) spanned by {ψ QR | µ ⊲ λ and Q, R ∈ Path h (µ)}.(4)The k-linear map * : H h (n, s) → H h (n, s) determined by (ψ ST ) * = ψ TS , for all λ ∈ P h (n) andall S, T ∈ Path h (λ), is an anti-isomorphism of H h (n, s).Given λ ∈ P h (n), the Specht module S s (λ) is the graded left H h (n, s)-module with basis {ψ S | S ∈ Path h (λ)}.The action of H h (n, s) on S s (λ) is given by 2 and m αβ = 3, we have the braid-commutativity relations The set {c ST | S, T ∈ Std + n,s (λ), λ ∈ P h (n, s)} is a k-basis of S h (n, s).(3)If S, T ∈ Std + n,s(λ), for some λ ∈ P h (n), and a ∈ S h (n, s) then there exist scalars r SU (a), which do not depend on T, such that ac ST = Definition 6.18.Given λ ∈ P h (n, s), the standard module ∆(λ) is the graded left S h (n, s)-module with basis {c S | S ∈ Std + n,s (λ)}.The action of S h (n, s) on ∆(λ) is given by ac S = U∈Std + n,s λ).The algebra S h (n, s) is quasi-hereditary with graded integral cellular basis{c S P λ c P λ T | S, T ∈ Std + n,s (λ), λ ∈ P h (n, s)}with respect to the Bruhat ordering on Λ h (n, s) and anti-involution * .That is, we have that (1) Each c ST is homogeneous of degree deg(c ST ) = deg(S) + deg(T), for λ ∈ P h (n, s) and S, T ∈ h is the k-submodule of S h (n, s) spanned by {c QR | Q, R ∈ Std + n,s (µ) for µ ⊲ λ}.(4)The k-linear map * : S h (n, s) → S h (n, s) determined by (c ST ) * = c TS , for all λ ∈ P h (n, s)and all S, T ∈ Std + n,s (λ), is an anti-isomorphism of S h (n, s).